Vanity Plate is a US term for a custom selected state license plate with numbers, letters, or a combination of the two. They are purchased from the state you live in and the funds go to state programs and expenditures, then it’s part of the annual Registration Fees for the car. It’s a form of self imposed state tax in exchange for a customized legal license plate for your car. It’s not required at all, completely optional. But what’s important is the plates are authorized by, made by, and distributed by the state.
I have no idea about outside the US. But that Personalized Number Plate website looks like it’s just a custom license plate for fun only, not a legally acceptable license plate. But in other countries automobile registration fees and license plates may be administered in a completely different way.